# Reciprocal moire lattice vectors
G = bandcalc.generate_twisted_lattice_by_shell(b, b, angle, 1)
GT = G[0]
GB = G[1]
GM = GT - GB
GM = np.array(sorted(GM, key=lambda x: np.abs(x.view(complex))))[1:]
GM = np.array(sorted(GM, key=lambda x: np.angle(x.view(complex))))
# Real space grid
size = np.linspace(-10, 10, 100)
grid = np.meshgrid(size, size)
# Results
fig, ax = plt.subplots()
moire_potential = bandcalc.calc_moire_potential_on_grid(grid, GM, Vj)
contour = bandcalc.plot_moire_potential(ax, grid, moire_potential, alpha=0.4)
twisted_lattice = bandcalc.generate_twisted_lattice_by_shell(a, a, angle, 15)
bandcalc.plot_lattice(ax, twisted_lattice[0], "r.")
bandcalc.plot_lattice(ax, twisted_lattice[1], "b.")
moire_lattice = bandcalc.generate_lattice_by_shell(m, 1)
bandcalc.plot_lattice(ax, moire_lattice, "ko")
ax.axis("scaled")
ax.set_xlim([size.min(), size.max()])
ax.set_ylim([size.min(), size.max()])
plt.colorbar(contour, label=r"$\Delta(\mathbf{r})$ in meV")
plt.show()
for i in range(7):
## Calculate the wannier function
R = 1 / 3 * (moire_lattice[5] + moire_lattice[3] +
moire_lattice[6]).astype(np.float32)
r = 5 * bandcalc.generate_monkhorst_pack_set(m, 80)
wannier_function = bandcalc.calc_wannier_function_gpu(hamiltonian,
k_points,
rec_moire_lattice,
r,
R,
band_index=i)
## Plot the results
bandcalc.plot_bandstructure(ax[i, 0],
np.real(bandstructure),
k_names,
"k",
alpha=0.3)
ax[i, 0].plot(np.real(sorted_bandstructure)[:, i], "k", lw=2)
ax[i, 1].tricontourf(r[:, 0], r[:, 1], np.abs(wannier_function), alpha=0.8)
lat = bandcalc.generate_lattice_by_shell(m, 3)
bandcalc.plot_lattice(ax[i, 1], lat, "ko")
ax[i, 1].plot(R[0], R[1], "ro")
for num, vec in enumerate(moire_lattice):
ax[i, 1].text(vec[0] + 0.2, vec[1], num, color="#999")
plt.tight_layout()
plt.savefig("wannier_all_bands.pdf")
energies, prefix = bandcalc.get_unit_prefix(
np.sort(
np.linalg.eigvals(
bandcalc.calc_hamiltonian(rec_moire_lattice, potential_matrix,
m)(k_point))))
energy_slice = energies[energy_level -
5 if energy_level - 5 > -1 else None:energy_level +
5 if energy_level + 5 < len(energies) else None]
energy = np.real(energies[energy_level])
#np.save("wavefunction_moire_{}deg_energy_level_{}_shells_8.npy".format(angle, energy_level), wavefunction)
# Plot the results
fig, axs = plt.subplots(nrows=1, ncols=2, figsize=(11, 5))
bandcalc.plot_lattice(axs[0], rec_moire_lattice, "ko")
axs[0].text(0.05,
0.95,
f"{angle}°",
transform=axs[0].transAxes,
va="top",
bbox={
"fc": "white",
"alpha": 0.2
})
axs[0].set_xlabel(r"nm$^{-1}$")
axs[0].set_ylabel(r"nm$^{-1}$")
bandcalc.plot_moire_potential(axs[1],
grid,
moire_potential,
# Real space moire potential
moire_potential = bandcalc.calc_moire_potential_on_grid(grid, GM, Vj)
# Reciprocal space moire potential
mp_moire = bandcalc.generate_monkhorst_pack_set(m, 100)
moire_potential_pointwise = bandcalc.calc_moire_potential(mp_moire, GM, Vj)
rec_moire_potential = bandcalc.calc_moire_potential_reciprocal_on_grid(
mp_moire, grid_r, moire_potential_pointwise)
# Results
fig, axs = plt.subplots(nrows=2, ncols=2)
contour = bandcalc.plot_moire_potential(axs[0,0], grid, np.real(moire_potential), alpha=0.4)
moire_lattice = bandcalc.generate_lattice_by_shell(m, 2)
bandcalc.plot_lattice(axs[0,0], moire_lattice, "ko")
vor_m = Voronoi(moire_lattice)
voronoi_plot_2d(vor_m, axs[0,0], show_points=False, show_vertices=False)
bandcalc.plot_lattice(axs[0,0], mp_moire, "b,", alpha=0.4)
rec_moire_lattice = bandcalc.generate_lattice_by_shell(rec_m, 3)
axs[0,0].axis("scaled")
axs[0,0].set_xlim([size.min(), size.max()])
axs[0,0].set_ylim([size.min(), size.max()])
for i, ax in enumerate(axs.ravel()[1:]):
bandcalc.plot_lattice(ax, rec_moire_lattice, "ko")
fun = [np.real, np.imag, np.abs][i]
title = ["real", "imaginary", "absolute"][i]
rec_contour = bandcalc.plot_moire_potential(ax, grid_r,
[0, 4 * np.pi / (3 * a)], # K
[0, 0], # Gamma
[2 * np.pi / (np.sqrt(3) * a), 0], # M
[2 * np.pi / (np.sqrt(3) * a), -2 * np.pi / (3 * a)]
]) # K
k_names = [r"K", r"$\Gamma$", r"M", r"K"]
path = bandcalc.generate_k_path(points, N)
# Calculate Brillouin zones
vor = Voronoi(lattice)
# Calculate band structure
potential_matrix = bandcalc.calc_potential_matrix(lattice)
hamiltonian = bandcalc.calc_hamiltonian(lattice, potential_matrix, m)
bandstructure, prefix = bandcalc.get_unit_prefix(
bandcalc.calc_bandstructure(path, hamiltonian))
# Plots
fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(11, 5))
bandcalc.plot_lattice(ax[0], lattice, "ko")
bandcalc.plot_k_path(ax[0], path, "r")
voronoi_plot_2d(vor, ax[0], show_points=False, show_vertices=False)
bandcalc.plot_bandstructure(ax[1], bandstructure, k_names, "k")
ax[1].set_ylabel(r"$E - \hbar\Omega_0$ in {}eV".format(prefix))
ax[1].set_ylim([0, 4])
plt.tight_layout()
plt.show()
import matplotlib.pyplot as plt from scipy.spatial import Voronoi, voronoi_plot_2d #pylint: disable=E0611 import bandcalc # Constants a = 1 N = 1000 # Reciprocal lattice vectors b1 = np.array([2 * np.pi / (np.sqrt(3) * a), 2 * np.pi / a]) b2 = np.array([2 * np.pi / (np.sqrt(3) * a), -2 * np.pi / a]) b = np.vstack([b1, b2]) lattice = bandcalc.generate_lattice_by_shell(b, 1) # Monkhorst-Pack lattice mp_lattice = bandcalc.generate_monkhorst_pack_set(b, 10) # Results fig, ax = plt.subplots() vor = Voronoi(lattice) voronoi_plot_2d(vor, ax, show_points=False, show_vertices=False) bandcalc.plot_lattice(ax, lattice, "ko", alpha=0.3) bandcalc.plot_lattice(ax, mp_lattice, "k.") ax.set_xlim([lattice[:, 0].min() - 2, lattice[:, 0].max() + 2]) ax.set_ylim([lattice[:, 1].min() - 2, lattice[:, 1].max() + 2]) plt.show()
# Real space moire lattice vectors
moire_lattice = bandcalc.generate_lattice_by_shell(m, shells)
# Wannier functions should be centered around potential minima
potential_shift = np.array([m[0, 0], m[0, 1]*1/3])
print(f"Potential shift: {potential_shift}")
R0 = potential_shift
R1 = m[0]+potential_shift
R2 = 2*m[0]+potential_shift
## Calculate the wannier function
R = R0
r = 4*bandcalc.generate_monkhorst_pack_set(m, 80)#+R
wannier_function = bandcalc.calc_wannier_function_gpu(
hamiltonian, k_points, rec_moire_lattice, r, R, band_index=energy_level)
## Plot the results
if mode == "3d":
bandcalc.plot_trisurface_3d(r[:,0], r[:,1], np.abs(wannier_function))
elif mode == "2d":
fig, ax = plt.subplots()
contour = ax.tricontourf(r[:,0], r[:,1], np.abs(wannier_function), alpha=0.8)
bandcalc.plot_lattice(ax, bandcalc.generate_lattice_by_shell(m, 4), "k.")
ax.plot(R[0], R[1], "ro")
ax.set_xlabel("nm")
ax.set_ylabel("nm")
plt.colorbar(mappable=contour)
plt.show()